5.1. Flow structure

Figure 4 shows the evolution of the LEV for the four cases summarized in table 1, using the $\lambda _{ci}$ (swirling strength) criterion (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). The phase-averaged flows are visualized using isosurfaces of normalized swirling strength (${\lambda ^2_{ci}}/{\textrm {max}\lambda ^2_{ci}}$) at values of 10 % and 15 %, following the guidelines given by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999). Plenoptic PIV data were acquired over the entire span for the $J_{3.25}=0.54$ case, and subsets of the span for the others, to provide the breadth of parameter space presented here. The $J_{3.25}=0.54$ case is hereafter referred to as the baseline case.

Figure 4.

The LEV formation shown with isosurfaces of normalized swirling strength levels of 0.10 and 0.25, for combinations of $J=0.54$ and 1.36 and $R_g/c$=3.25 and 2.5, with $\alpha _{eff} = 33^{\circ }$. Each row depicts a different case; (a–e) $J_{3.25}=0.54$, (f–j) $J_{2.5}=0.54$, (k–o) $J_{3.25}=1.36$ and (p–t) $J_{2.5}=1.36$. Each column represents a specific roll angle comprising, from left to right, $\phi =12^{\circ }$, $17^{\circ }$, $21^{\circ }$, $30^{\circ }$ and $40^{\circ }$.

It should be noted that the wing mounting bracket, which thickens the wing along the root, will have some influence on the development of the LEV system. However, based on the visualization of the baseline case, it appears that the bracket has not substantially interfered with the formation of a conical LEV on the inboard portion of the wing.

Both of the $J=0.54$ cases ($R_g/c=3.25$ and 2.5) exhibit an attached LEV on the inboard region of the wing, for the duration of the measurements, consistent with the observations of Bross et al. (Reference Bross, Ozen and Rockwell2013), for a similar advance ratio. The highest advance ratio investigated in that study was $J=0.537$, and the authors noted similar LEV development for lower advance ratios. For the baseline case, shown in figures 4(a) through 4(e), the inboard region is captured within the plenoptic imaging domain; whereas, for the wing with $R_g/c=2.5$, it is inferred because the most inboard portion of the imaged vortex remains stationary. This observation was also supported by dye visualizations (not shown).

Furthermore, for the baseline case, three distinct behaviours can be observed across the span, as shown in figure 5. Region 1 ($z/b\lessapprox 0.37$) is characterized by the attached, and essentially stationary LEV for the duration of the starting roll manoeuvre. In region 2 ($0.37 \lessapprox z/b \lessapprox 0.74$), the LEV separates from the wing, forming an arch structure, which sheds and is followed by the creation of subsequent vortices. By $\phi =30^{\circ }$, a second LEV structure is apparent, and a third is evident by $\phi =40^{\circ }$. In region 3 ($z/b \gtrapprox 0.74$), tip effects dominate, and the LEV structure reattaches and is pinned to the leading edge of the wing. A tip vortex is also formed.

Figure 5.

Three distinct topological regions shown in the baseline case.

For the cases with $R_g/c=2.5$ (figures 4f through 4j and 4p through 4t), data were acquired only along the central region of the wing ($0.35\leqslant z/b \leqslant 0.75$). The recorded region of the $J_{2.5}=0.54$ case also appears to support the existence of the two inboard regions (region 3 is not captured in the domain); however, unlike the baseline case, which exhibits sharp transitions between the three spanwise regions, the LEV structure transitions smoothly from the inboard attached structure to the separating structure at the mid-span (which is presumed to be a portion of an arch vortex displaced somewhat further outboard in comparison with the baseline case). The transition to a separating structure is most evident at $\phi =30^{\circ }$; whereas, it appears that, by $\phi =40^{\circ }$ the first LEV structure has advected downstream and dissipated as it is replaced by a second structure.

At $\phi =12^{\circ }$, the $J=1.36$ cases exhibit structures similar to the $J=0.54$ cases at later times in their evolution. The measured flow structure for $J_{3.25}=1.36$ in figure 4(k) exhibits strong similarities to the $J_{3.25}=0.54$ case at $\phi =21^{\circ }$ (figure 4c). Similarly, the central region of the $J_{2.5}=1.36$ case, visible in figure 4(p), bears a strong resemblance to the $J_{2.5}=0.54$ case at $\phi =21^{\circ }$. Such observations of vortex-evolution delays governed by advance ratio have previously been made by Bross et al. (Reference Bross, Ozen and Rockwell2013) and Johnson et al. (Reference Johnson, Thurow, Wabick, Berdon and Buchholz2020).

5.2. Circulation distributions

Figure 6 shows the variation in LEV circulation – measured within the control region illustrated in figure 3 – for the four rolling-wing cases. Circulation values were measured at $\phi =6^{\circ }$, $12^{\circ }$, $17^{\circ }$, $21^{\circ }$, $26^{\circ }$, $30^{\circ }$, $40^{\circ }$ and $49^{\circ }$. at spanwise increments of 0.066$c$, and interpolated by a cubic spline using MATLAB's interp2 function on each spanwise plane. Approximate boundaries between regions 1, 2 and 3 – to the extent that they are included within the domains visualized in figure 6 – are depicted as dashed lines. All three regions are shown for $J_{3.25}=0.54$, whereas only regions 1 and 2 are evident for $J_{2.5}=0.54$ and $J_{2.5}=1.36$. Regions 2 and 3 are depicted for $J_{3.25}=1.36$. The boundary appears to move inboard for $J_{2.5}=1.36$; hence the sloped line in figure 6(d).

Figure 6.

Spatial and temporal distribution of dimensionless circulation ($\varGamma /(U_{\infty } c)$). The approximate boundaries between the regions defined in figure 5 are depicted here using dashed lines. In (a) $J_{3.25}=0.54$, all three regions are shown; in (b) $J_{2.5}=0.54$, regions 1 and 2 are shown; in (c) $J{3.25}=1.36$, regions 2 and 3 are shown; and in (d) $J{2.5}=1.36$, regions 1 and 2 are shown. Note the different spatial and circulation scales in each plot.

The three regions defined in § 5.1 exhibit distinct patterns of circulation growth tied to the disparate patterns of flow evolution found in these regions. For the baseline case, the circulation within region 1 grows slowly throughout the stroke (figure 6a). A striking transition in circulation is observed at the boundary between regions 1 and 2. In region 2, the maximum circulation observed is approximately double that in region 1, and exhibits two notable peaks at approximately $\phi = 21^{\circ }$ and $30^{\circ }$, associated with the development of the second and third vortices. In contrast, in region 3, the circulation grows monotonically and remains relatively constant after approximately $\phi =25^\circ$.

It is not unexpected that the vortex strength within region 1 should be less than that further outboard, due to the lower velocity of the wing at inboard locations, and three-dimensional effects, which are further examined in § 5.4. Also, as discussed in § 2, it has been widely hypothesized, and demonstrated in some cases, that the conical vortex structure is stabilized by transport mechanisms that act to limit the circulation. Indeed, Jardin (Reference Jardin2017) observed, for a large-aspect-ratio revolving wing in quiescent flow, that the evolution in normal force was well represented by the sectional normal force at around 80 % span, demonstrating that a conical LEV attached to the inboard portion of the wing does not necessarily contribute a significant fraction of the lift.

Consistent with the observed flow structure behaviour, reducing the radius of gyration appears to decrease unsteadiness in the circulation, as the $J_{2.5}=0.54$ case (figure 6b) exhibits more uniform circulation growth in time within the mid-span region of the wing. Similarly, figures 4(f) through 4(j) suggest that the vortex growth and shedding events occur over greater roll displacements. As with the baseline case, a sharp increase in circulation develops at approximately $z/b=0.47$, suggesting the transition from region 1 to region 2 delineated by the dashed line in figure 6(b), despite an apparently continuous variation in flow structure. It is also noteworthy that the location of this transition is further outboard on the wing than for the baseline case, consistent with the spanwise shift of LEV features observed in figure 4. While the tip region is not included in the $J_{2.5}=0.54$ measurement domain, it appears that the transition to region 3 would also occur further outboard.

For the $J=1.36$ cases, the circulation distribution is aliased in time due to the rapid shedding of vortices from the leading edge. However, the data suggest spanwise variation in circulation that is roughly consistent with the fast-roll cases, despite that the measured values are somewhat arbitrary in these cases due to the advection of vorticity out of the downstream boundary. The spatial and temporal evolution of circulation for the $J_{2.5}=1.36$ case bears a strong qualitative similarity to $J_{2.5}=0.54$. However, an important distinction between the $J_{2.5}=0.54$ and $J_{2.5}=1.36$ cases is that, whereas the boundary dividing the low- and high-circulation regions (ostensibly regions 1 and 2) is stationary in the former, it moves inboard for the slow-roll ($J=1.36$) case as the motion progresses, as indicated by the slanted dashed line in figure 6(d). This indicates a destabilization of the inboard LEV in the slow-roll case, as would be expected for a translating wing in contrast to the fast-roll cases, where the LEV remains attached over the inboard region of the wing. This suggests that the existence of region 1 and its associated characteristics may not be caused by rotational effects, but merely prolonged by them. Data for the $J_{3.25}=1.36$ case are acquired to the tip of the wing, revealing a transition from region 2 to region 3 at approximately $R_g/c=0.7$, marked by the rapid drop in circulation. The location of this boundary is comparable to the baseline case, which indicates that the transition from region 2 to region 3 is due, primarily to tip effects, and not rotational effects.

5.3. Circulation budget closure and non-inertial contributions

Closure of the circulation budget described by (4.1) implies that the salient transport processes are adequately modelled, and that the measurements and derived quantities are sufficiently accurate. To assess the implementation of (4.1), spatial derivatives were evaluated using second-order central differencing of the plenoptic PIV velocity fields, and integrations were performed using the trapezoidal rule. The integral on boundary 4 of the control region was evaluated using pressure measurements acquired at $z/b=0.5$ and chordwise positions of $x/c=0.015$ and $x/c=0.65$, employing the fundamental theorem of calculus.

The terms on the right and left sides of (4.1), are plotted in figure 7 for $J_{3.25}=0.54$ at the mid-span ($z/b=0.50$). The left side, $\mathrm {d}\varGamma /\mathrm {d}t$, is compared with the sum of the terms on the right side (designated ‘integrated vorticity flux’ in figure 7. Each of the terms on the right side of (4.1) are designated, generally, as instantaneous integrated vorticity fluxes (IIVF). The shear-layer flux through boundary 1 of the control region is labelled ‘shear’; whereas the combined contributions of tilting and Coriolis tilting are designated, ‘tilt and Coriolis’. The other terms on the right side of (4.1) include the spanwise convective flux and the diffusive flux through boundary 4. The computed rate of change of circulation ($\mathrm {d}\varGamma /\mathrm {d}t$) agrees well with the sum of the terms on the right-hand side of (4.1), indicating that the transport mechanisms governing control-region circulation are adequately described by (4.1). Based on this agreement, for other planes and cases, the diffusive flux on boundary 4 will be computed directly from (4.1), using plenoptic PIV data to obtain the remaining terms, and assuming equality.

Figure 7.

Vorticity flux distribution for $J_{3.25}=0.54$ at $z/b=0.5$.

Figure 7 shows that the dominant contributions to the circulation within the control region at $z/b=0.5$ are the leading-edge shear layer and the surface diffusive flux on the surface of the plate. The shear-layer contributions in all cases are composed solely of measurements of the velocity field in the non-inertial frame, and do not include the Coriolis contributions. However, the distinction is practically unimportant in this case since the Coriolis contribution to the in-plane convective flux is negligible (i.e. $2\varOmega \ll \text {typical}$ values of $\omega _z$ in the shear layer). It should be noted that the sign of the vorticity in the shear layer and LEV is primarily negative; hence the negative circulation contribution of the shear layer.

The diffusive flux is opposite in sign, and thus strongly regulates the total circulation as the shear-layer contribution continues throughout the stroke. Notably, the contributions due to the tilting and Coriolis terms remain small in comparison with those of the dominant fluxes. Their negligible contribution to the circulation budget will later be shown to be robust across other spanwise locations and kinematic cases.

Figure 8 further examines the $x$- and $y$-components of the physical and Coriolis-tilting contributions. Figure 8(a) shows that the spatial integral of $x$- and $y$-tilting terms (i.e. the separate integrals of $\omega _x \partial u_z/\partial x$ and $\omega _y \partial u_z/\partial y$, respectively) within the control region exhibit a highly symmetric behaviour as the roll angle progresses, resulting in substantial cancellation of their contributions to the circulation within the control region. This was also observed by Wojcik & Buchholz (Reference Wojcik and Buchholz2014b) for a wing revolving in quiescent fluid. The Coriolis tilting, shown in figure 8(b), behaves similarly. The net contributions of each of the tilting and Coriolis terms, as well as their sum is presented in figure 8(c). It is evident that the magnitudes of the net Coriolis and physical tilting contributions are indistinguishable within the measurement uncertainties.

Figure 8.

Coriolis and tilting fluxes, integrated over the control region, for $J_{3.25}=0.54$ at $z/b=0.5$; (a) $x$- and $y$-tilting terms ($\omega _x \partial u_z/\partial x$, and $\omega _y \partial u_z/\partial y$), (b) corresponding Coriolis terms ($2\varOmega _x \partial u_z/\partial x$, and $2\varOmega _y \partial u_z/\partial y$) and (c) a comparison between tilting and Coriolis contributions, as well as the net values of combined tilting and Coriolis contributions.

5.4. Vorticity transport analysis on the baseline ($J_{3.25}=0.54$) case

In this section, we will develop an understanding of the transport mechanisms governing the evolution of the LEV system circulation by considering the baseline case in detail. Figure 9 presents the contributions to the circulation budgets at nine spanwise locations on the wing. The data for $z/b=0.20$ to 0.30 (figure 9a–c) reside in region 1; $z/b=0.40$ to 0.60 (figure 9d–f) reside in region 2; and those for $z/b=0.80$ to 0.90 (figure 9g–i) reside in region 3. Note that figure 9(e) contains data presented in figure 7. Data are presented as dimensionless vorticity fluxes ($IIVF/c U_{\infty }\varOmega _x$).

Figure 9.

Vorticity transport budget for $J_{3.25}=0.54$. Data are shown for region 1 (top row; (a) $z/b=0.20$, (b) $z/b=0.25$ and (c) $z/b=0.30$), region 2 (middle row; (d) $z/b=0.40$, (e) $z/b=0.50$ and (f) $z/b=0.60$) and region 3 (bottom row; (g) $z/b=0.80$, (h) $z/b=0.85$ and (i) $z/b=0.90$).

Justified by the circulation budget closure demonstrated in § 5.3, the diffusive-flux contribution, from the surface of the wing is inferred from (4.1) after computing the values of the other terms. Since the in-plane convective fluxes through control-region boundaries 2 (top) and 3 (downstream) are negligible in the present case, the shear-layer flux through boundary 1 is the main contribution to the in-plane convective term, and is labelled, shear layer. The tilting term is computed according to (4.4), and combines the physical and Coriolis tilting contributions.

At each of the spanwise locations represented in figure 9, the rate of circulation growth $\mathrm {d} \varGamma / \mathrm {d}t$ is generally negative (LEV strength increasing in time) and significantly smaller than the magnitudes of the dominant flux contributions in each case, reflecting that circulation growth within the control region is the result of competing sources and sinks of vorticity. However, the transport processes exhibit striking differences in each of the three regions. The distinct transport characteristics of each region are examined in detail below.

5.4.1. Region 1

The shear-layer flux is the source that drives the growth of the LEV. In region 1 (figure 9a–c), the shear-layer contribution is substantially balanced by the diffusive flux from the wing surface early in the rotation. However, with increasing roll angle, the diffusive flux diminishes, and is replaced by a monotonically growing spanwise convective flux as centrifugal pumping on the wing surface begins to develop. As noted in § 2, some prior studies have measured a significant spanwise convective flux on revolving wings (e.g. Wong et al. Reference Wong, laBastide and Rival2017) while others have not (Wojcik & Buchholz Reference Wojcik and Buchholz2014b; Onoue & Breuer Reference Onoue and Breuer2017).

Thus, the region 1 data illustrate that the importance of the spanwise convective flux is time dependent. For a wing of aspect ratio 3 and root cut-out of 1 chord length, rotating in quiescent fluid, Chen et al. (Reference Chen, Wu and Cheng2019) found that spanwise convection was not significant until after approximately $22^{\circ }$ of rotation, but diminished again after about $90^{\circ }$ of rotation. In the present experiments, the growth of the spanwise convective flux appears to have similar timing; however, there was not sufficient displacement to investigate whether spanwise convection diminished in the longer term. It should be noted that Chen et al. (Reference Chen, Wang, Zhou, Wu and Cheng2022) found that the importance of spanwise convective flux diminished significantly when Reynolds number reduced from $Re_c=1500$ to 100.

To understand the physical processes that govern the growth of the spanwise convective flux in region 1, figure 10 shows the distribution of spanwise vorticity, as well as the integrand of spanwise convective flux ($u_z \partial \omega _z / \partial z$) and its constituent quantities. These include spanwise velocity ($u_z$) and the derivative of the vorticity in the spanwise direction ($\partial \omega _z / \partial z$), for $z/b=0.3$. With increasing progression in time, figures 10(a) through 10(d) show that the vortex becomes more diffuse, and its development begins to regress; descending toward the surface, reattaching to the downstream boundary layer, and decreasing in size. The start of this regression is concurrent with appearance of a strong outboard flow in the core of the vortex, evident in figures 10(f) and 10(g), and concomitant growth of the integrated spanwise convective flux observed in figure 9(c).

Figure 10.

The LEV formation and components of the spanwise convective flux (spcf) at $z/b=0.3$. Panels (a) to (d) depict the LEV evolution of the vorticity field. Panels (e) to (h) show the spanwise velocity distributions. Panels (i) to (l) show distributions of $\partial \omega _z / \partial z$. Finally, panels (m) to (p) show the spanwise convective flux distributions ($u_z \partial \omega _z / \partial z$). The columns represent roll angles of $\phi =17^{\circ }$, $21^{\circ }$, $26^{\circ }$ and $30^{\circ }$ from left to right.

Notably, the spanwise flow in the LEV core also diminishes abruptly between $\phi =26^{\circ }$ and $30^{\circ }$ (figure 10g,h). However, as figure 9(c) shows, the integral contribution of the spanwise convective flux does not diminish, but merely persists with a modest growth rate. This is understood by considering the evolution of the vorticity gradient in figures 10(i) through 10(l), which grows monotonically, within the LEV, throughout the manoeuvre. Significant values of $\partial \omega _z / \partial z$ develop not only in the core of the LEV, but also in the attached boundary layer downstream of the control region ($x>0.52$). Despite the complex variability of the spanwise velocity, the spanwise vorticity gradient is correspondingly modulated in order to maintain a smoothly varying spanwise convective flux that mitigates the decline of the diffusive flux. This suggests that the transport features observed here are driven by constraints imposed by the global flow dynamics.

5.4.2. Region 2

In contrast to region 1, the spanwise convective flux is relatively insignificant throughout the roll motion in region 2 (figure 9d–f), and the circulation budget is dominated by a balance between the shear-layer and surface-diffusive fluxes. This behaviour is characteristic of planar motion of a nominally two-dimensional wing (e.g. Shih & Ho Reference Shih and Ho1994; Eslam Panah et al. Reference Eslam Panah, Akkala and Buchholz2015).

A brief reduction in the magnitudes of both the shear-layer and diffusive fluxes occurs around $\phi =26^{\circ }$. To better understand the cause of this fluctuation, figure 11 shows the corresponding evolution of the spanwise vorticity field at $z/b=0.5$. At $\phi =26^{\circ }$ (figure 11c), the phase-averaged LEV begins to become diffuse, and the region of secondary vorticity, which has grown in size during the manoeuvre, starts to weaken and atrophy. The atrophy is likely due to entrainment of the secondary vorticity layer into the shear layer, as observed by Eslam Panah et al. (Reference Eslam Panah, Akkala and Buchholz2015). The existence of a significant accumulation of secondary vorticity is in distinct contrast to region 1 where, as figure 10 shows, it is essentially non-existent. A second LEV is forming at $\phi =30^{\circ }$ and $z/b=0.50$ (figure 11d) where figure 9(e) shows that the shear-layer flux has recovered.

Figure 11.

Spanwise vorticity ($\omega _z$) at $z/b=0.5$, (a) $\phi =17^{\circ }$, (b) $\phi =21^{\circ }$, (c) $\phi =26^{\circ }$, (d) $\phi =30^{\circ }$, (e) $\phi =40^{\circ }$ and (f) $\phi =49^{\circ }$.

5.4.3. Region 3

The flow field is highly three-dimensional in region 3, and exhibits flow evolution and flux contributions that are very different from regions 1 and 2. As noted in § 5.2, region 3 is characterized by tip effects and appears to be relatively insensitive to rotation. As figures 12(a) through 12(d) show, over time, the vortex evolves into a rapidly reattaching shear layer, suggesting that a strong favourable pressure gradient develops late in the manoeuvre. This is supported by the vorticity transport budgets for region 3, illustrated in figures 9(g) through 9(i), which show a strong diffusive contribution of negative vorticity by approximately $\phi =40^{\circ }$. Until approximately $\phi = 25^{\circ }$, the shear layer is the primary source of LEV circulation. Its eventual weakening and concomitant favourable pressure gradient are plausibly connected with the development of the tip vortex.

Figure 12.

The LEV formation and components of the spanwise convective flux (spcf) at $z/b=0.9$. Panels (a) to (d) depict the evolution of the vorticity field. Panels (e) to (h) show the spanwise velocity distributions. Panels (i) to (l) show distributions of $\partial \omega _z / \partial z$. Finally, panels (m) to (p), show the spanwise convective flux distributions ($u_z \partial \omega _z / \partial z$). The columns represent roll angles of $\phi =21^{\circ }$, $26^{\circ }$, $30^{\circ }$ and $40^{\circ }$ from left to right.

The spanwise convective flux also evolves concurrently with the shear-layer and diffusive fluxes. There is a characteristic drop in its magnitude around $\phi =25$ to $30^{\circ }$, which corresponds to the drop in diffusive flux observed in region 2 during the formation of the second LEV, again suggesting that the three regions do not evolve independently. Similarly to the diffusive contribution in region 2, the spanwise convective contribution in region 3 subsequently recovers.

As with region 1, the persistent contribution of the spanwise convective flux can be further understood by examining its spatial distribution, as well as that of its constituents, as shown in figure 12 for $z/b=0.90$. At $\phi =21^{\circ }$ (figure 12e), the spanwise flow in the vicinity of the LEV and adjacent boundary layer is inboard (blue). An inboard flow persists in the shear layer near the leading edge, throughout the manoeuvre, whereas a region of outboard spanwise flow is apparent at $\phi =26^{\circ }$, and grows in size with increasing roll angle. The outboard flow is predominantly located near the wing surface, and is likely driven, in part, by the tip vortex. Interestingly, the distribution of spanwise vorticity gradient is roughly complementary to the spanwise flow (i.e. positive values of $\partial \omega _z / \partial z$ are approximately coincident with negative values of $u_z$, and vice versa), such that the spanwise convective flux produces positive contributions to the net circulation within the shear layer, and adjacent to the wing surface. Similar to region 1, the patterns of spanwise flow and vorticity gradient are quite complex.

5.5. The role of radius of gyration and advance ratio

The previous section established characteristic patterns of vorticity transport in region 1, where an attached LEV was observed and rotational accelerations were important, and in region 2, where rotational accelerations appeared to be relatively unimportant. Lentink & Dickinson (Reference Lentink and Dickinson2009a) showed that the influence of rotational accelerations is inversely related to the Rossby number, which they expressed as $Ro=U^2/(\varOmega ^2 R_g c)$ for a wing or blade spinning in a free stream. In terms of the advance ratio, this may be expressed as $Ro = [(R_g/c)(1 + J^2)]^{-1}$, such that the effects of rotational accelerations are reduced when both radius of gyration ($R_g/c$) and advance ratio are increased. Thus, as either parameter increases, we might expect the vorticity transport characteristics to tend toward region 2 behaviour and, conversely, with decreasing parameter values, transport flux behaviours characteristic of region 1 might become more prominent. In this section, we test this hypothesis by considering the three additional cases listed in table 1, in which $J$ and $R_g/c$ are varied.

Figure 13 shows the vorticity transport budget for the $J_{2.5}=0.54$ case at $z/b=0.35$ and 0.60. These spanwise locations are ostensibly within regions 1 and 2, respectively, considering the outboard shift of LEV features in comparison with the baseline case. Figure 13(a) shows that the primary transport attributes of region 1 are robust as radius of gyration is reduced. Specifically, the diffusive flux forms a significant contribution to the overall budget early in the manoeuvre, but is diminished later, as the spanwise convective flux increases. Again, tilting fluxes (physical and Coriolis) are negligible. The visualizations in figure 4 indicate diminished unsteadiness and complexity in the mid-span region, for the $J_{2.5}=0.54$ case, in comparison with the baseline case. Similarly, the evolution of the vorticity fluxes at $z/b=0.60$ (figure 13b) lacks the significant fluctuations in shear-layer and diffusive fluxes that are evident in the baseline case (figure 9d–f). A notable difference for $J_{2.5}=0.54$ is that, with the exception of the early part of the motion (i.e. $\phi <15^{\circ }$), the spanwise convective flux – while exhibiting oscillations, as in the baseline case – is generally of similar magnitude to the diffusive flux in the mid-span region of the wing (figure 13b). While this was also observed in the outboard portion of region 2 for the baseline case (figure 9f), the shear-layer and diffusive fluxes were generally dominant over the majority of region 2. Thus, with reduction of the radius of gyration, the distinction between regions 1 and 2 appears to be somewhat obscured, as spanwise transport remains significant in the mid-span region for $J_{2.5}=0.54$, supporting our hypothesis on the effect of reducing radius of gyration. Furthermore, the correlation between spanwise convective transport and the relative stability of the LEV persists in this region.

Figure 13.

The vorticity transport budget for $J_{2.5}=0.54$, at (a) $z/b=0.35$, and (b) $z/b=0.60$.

The effects of advance ratio can be observed by comparing these data with figure 14, which depicts the temporal evolution of the flux contributions in the slow-roll cases at the same spanwise locations ($z/b=0.35$ and 0.60). Figures 14(a) and 14(b) contain data for the $J_{2.5}=1.36$ case, and figures 14(c) and 14(d) contain data for the $J_{3.25}=1.36$ case.

Figure 14.

The vorticity transport budgets for $J_{2.5}=1.36$, at (a) $z/b=0.35$, and (b) $z/b=0.60$, and for $J_{3.25}=1.36$ and (c) $z/b=0.35$ and (d) $z/b=0.60$.

In all slow-roll cases, the vorticity transport budget is dominated by the shear-layer flux and the surface-diffusive flux which, unsurprisingly, resembles region 2 of the baseline case, for the spanwise locations shown in figure 14. In particular, the spanwise convective flux is typically negligible at $z/b=0.35$ and exhibits only modest fluctuations at $z/b=0.60$. However, the effects of advance ratio, as well as radius of gyration and spanwise location, on the magnitudes of the shear-layer and diffusive fluxes are quite significant. The dimensionless vorticity fluxes are, in general, significantly larger at $J=1.36$ than for $J=0.54$, with the exception of the $J_{2.5}=1.36$ case at $z/b=0.60$. It is not clear why the flux values are lower in that case.

Figure 15 shows the spatial and temporal evolution of the shear-layer contribution over the spanwise regions measured for each of the four cases listed in table 1. Again, the temporal resolution of the measurements is not sufficient to resolve all of the events in the evolution of the LEV (especially for the $J=1.36$ cases, which exhibit rapid shedding from the leading edge). However, the data are adequate to elucidate important spanwise variations in the shear-layer flux.

Figure 15.

Spatial and temporal evolution of the shear-layer flux contributions ($IIVF/U_{\infty }^2$). Dashed lines, representing boundaries between regions 1, 2 and 3, are copied from figure 6. Note the different spatial and flux-contribution scales in each panel; (a) $J_{3.25}=0.54$, (b) $J_{2.5}=0.54$, (c) $J_{3.25}=1.36$, (d) $J_{2.5}=1.36$.

Notably, all four cases exhibit maximum shear-layer flux at approximately $z/b=0.60$ around the initiation of the roll motion, despite having highly varied shear-layer flux distributions later in the manoeuvre. It is particularly noteworthy that the region of initially strong shear-layer contribution appears to reside exclusively in region 2 for the $J_{3.25}=0.54$, $J_{2.5}=0.54$ and $J_{3.25}=1.36$ cases, and possibly also the $J_{2.5}=1.36$ case. The shear-layer evolution for the baseline case, shown in figure 15(a), displays strong vertical banding, with sharp spanwise gradients evident at the boundaries between regions 1, 2 and 3, throughout much of the manoeuvre. In contrast, the boundaries are not as evident for the cases with $R_g/c=2.5$. This is consistent with earlier observations of flow structure, and that the vorticity flux behaviours are not as distinct at $z/b=0.35$ and 0.60 for the $J_{2.5}=0.54$ case and, as noted above, the shear-layer flux is significantly lower for the $J_{2.5}=1.36$ case than for the others. A strong transition is evident for the $J_{3.25}=1.36$ case at $z/b=0.70$, which coincides with the boundary between regions 2 and 3, and persists throughout the motion. The boundary between regions 2 and 3 is ostensibly outside the measurement volume for the $J_{2.5}=1.36$ case.

The difference in shear-layer strength between figures 15(c) and 15(d) is somewhat surprising because, for the larger $J$ value, rotational effects otherwise appear to be minimal, suggesting that radius of gyration would be relatively unimportant. Since, as discussed above, we expect stronger rotational effects for the $R_g/c=2.5$ cases, we expect that the shear-layer flux distribution for the $J_{3.25}=1.36$ case is more representative of a translating wing, and with further increases in $J$, the shear-layer flux patterns for $R_g/c=2.5$ might begin to look more like the $J_{3.25}=1.36$ case.